2 8 SECTION II. ADDITION. 38. If you have 5 cents and 3 cents and 2 cents, and count them together, how many cents do you find there are ? Counting them together, you find there are 10 cents. 39. The process of counting numbers together is addition. 40. The result found by addition is the sum or amount of the numbers added. Thus, the sum of 5 and 3 and 2 is 10. 41. The addition of numbers is indicated by the sign +, which is read plus. The sign = indicates equality, and is read equals, or is equal to. Thus, the expression 5 + 3 + 2 = 10 means that the sum of 5 and 3 and 2 is 10, and is read “five plus three plus two equals ten.” 42. Oral Exercises. 1. Name the sums of the pairs of numbers expressed below till you can give them rapidly at sight: m. By twos, beginning with 2; with 1. s, or neans plus below III. Add the numbers expressed in the following columns: (1.) (3.) (4.) (5.) (6.) (8.) 2 9 4 4 6 60 600 6000 3 6 2 3 3 30 300 3000 5 3 6 7 8 600 9000 7 2 8 4 8 80 800 8000 6 5 8 5 3 30 400 6000 3 8 8 5 50 500 5000 9 4 2 9 3 30 300 3000 1 7 9 700 9000 80 90 Begin at the bottom and add upward, naming only the results. Thus, in the first column, say 1, 10, 13, 19, 26, 31, 34, 36; sum, 36. Now, to see if you are right, begin at the WRITTEN WORK. Sum, 649 top and add downward. Thus, 2, 5, 10, 17, 23, 26, 35, 36; sum again, 36. Practise exercises of this kind till you can add with great rapidity. For further drill of this sort the teacher is referred to exercises on pages 59 and 61. Examples for the Slate. 43. ILLUSTRATIVE EXAMPLE I. What is the sum of 413, 102, and 134 ? Explanation. — To find the sum of large numbers 413 like these, we add the units, the tens, and the hun dreds separately ; hence, for convenience, we write 102 the numbers so that units of the same order may 134 be expressed in the same column. (Art. 6.) Drawing a line beneath, and adding the units (thus, 4, 6, 9), we find there are 9 units, which we write under the line in the units' place. Adding the tens (thus, 3, 4), we find there are 4 tens, which we write under the line in the tens' place. Adding the hundreds (thus, 1, 2, 6), we find there are 6 hundreds, which we write under the line in the hundreds place. The sum is, then, 6 hundreds 4 tens and 9 units, or 649. 44. ILLUSTRATIVE EXAMPLE II. What is the sum of 960, 748, 932, and 867 ? Explanation. — Writing the numbers as before, 960 and adding the units (thus, 7, 9, 17), we find there 748 are 17 units, which are equal to 1 ten and 7 units. We write the 7 units in the units' place, but keep 932 the 1 ten to add with the tens expressed in the next 867 column. Adding the tens (thus, 1, 7, 10, 14, 20), we find there are 20 tens, which are equal to 2 hundreds and no tens. We write () in the tens' place, to show there are no tens in the sum, but keep the 2 hundreds to add with the hundreds expressed in the next column. Adding the hundreds (thus, 2, 10, 19, 26, 35), we find there are 35 hundreds, which are equal to 3 thousands and 5 hundreds. We write 5 in the hundreds' and 3 in the thousands' place. The sum is, then, 3 thousands 5 hundreds 0 tens 7 units, or 3507. WRITTEN WORK. Sum, 3507 Keeping a number and adding it with the numbers expressed in the next column is called carrying. In working examples, use as few words as possible. Thus, in the above example, say merely, “7, 9, 17;* 1, 7, 10, 14, 20; 2, 10, 19, 26, 35; sum, 3507.” 1. Add together 6234, 785, and 5861. Addition of Decimals. WRITTEN WORK. 45. ILLUSTRATIVE EXAMPLE III. What is the sum of 425.37, 433.126, 0.076, 442.09, 0.6, and 0.319 ? Explanation. — Writing the numbers so that units of the same order may be expressed in the 425.37 same column, we begin with the units of the low433.126 est order (in this case thousandths) to add, and 0.076 proceed in the manner already explained, briefly 442.09 thus : thousandths, 9, 15, 21 ; write 1, carry 2 ; 0.6 hundredths, 2, 3, 12, 19, 21, 28; write 8, carry 2; 0.319 tenths, 2, 5, 11, 12, 15; write 5, carry 1; units, 1, 3, 6, 11; write 1, carry 1; tens, 1, 5, 8, 10; Sum, 1301.581 write 0, carry 1; hundreds, 1, 5, 9, 13; write 13; sum, 1301.581. 7. Add together 90.7, 43.68, 0.045, and 0.812. 8. Add together 0.005, 2.864, 0.9, and 0.25. 9. Add together forty-two thousandths, one hundred seventeen thousandths, thirteen and twenty-two hundredths, seven and five hundredths. * Do not stop to say “write 7 and carry 1,” but do it. 46. From the preceding examples we may derive the following Rule for Addition. 1. Write the numbers to be added so that units of the same order may be expressed in the same column. Draw a line beneath. 2. Add the units of each order separately, beginning with those of the loweste order. 3. When the sum of the units of any order is less than ten, write it under the line in its proper place; when ten or more, write only the units of the sum, and carry the tens to the numbers expressed in the next column. 4. Write the whole sum of the last addition. Proof. Repeat the work, adding downward instead of upward. WRITTEN WORK. Adding two or more columns at once. 47. Accountants often add at once the numbers expressed in two, three, or more columns. The following example will illustrate the method : Explanation. — Beginning with 29, add to it first 35 the 4 tens and then the 2 units of 42; then to the 72 sum the 8 tens and the 7 units of 87; and so on. 53 Naming the results merely, say 29, 69, 71; 151, 158; 188, 192; 242, 245; 315, 317; 347, 352. Adi34 ing downwards, say 35, 105, 107; 157, 160; 190, 194; 274, 281; 321, 323; 343, 352. 42 After practice it will be found unnecessary 29 to name all the results; and it is by omitting to name them that great rapidity is acquired. NOTE. The examples on the opposite page embrace the chief varieties in form of examples in Addition. After performing these, and before taking the Applications on page 18, pupils will usually need additional practice in similar work. Examples for such practice will be found on pages 59–63. 87 Sum, 352 |